- Thread starter
- Moderator
- #1
- Jan 26, 2012
- 995
Here's this week's problem.
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Problem: Use contour integration to show that
\[\int_{-\infty}^{\infty}\frac{e^{-2\pi i x\xi}}{(1+x^2)^2}\,dx = \frac{\pi}{2}(1+2\pi|\xi|)e^{-2\pi|\xi|} \]
for all $\xi$ real.
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Hint:
Remember to read the POTW submission guidelines to find out how to submit your answers!
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Problem: Use contour integration to show that
\[\int_{-\infty}^{\infty}\frac{e^{-2\pi i x\xi}}{(1+x^2)^2}\,dx = \frac{\pi}{2}(1+2\pi|\xi|)e^{-2\pi|\xi|} \]
for all $\xi$ real.
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Hint:
WLOG, suppose that $\xi\geq 0$ (this way, you don't have to worry about the absolute values for the time being). Then consider using the lower half circle as the contour for this integral.
Remember to read the POTW submission guidelines to find out how to submit your answers!